A device for the milling of longitudinally extending areas, particularly of the edges of continuous metal strips or long rods, is known from German Published Application DE-AS No. 15 02 100, where the axis of a cutter head equipped with hard alloy cutters is inclined by an angle .gamma. in relation to the plane of the strip to be trimmed. The hard alloy cutters are also disposed inclined by the same angle .gamma. in relation to the normal plane of the cutter head. However, in this known device the individual cutting edges are formed in a straight line and by themselves cannot define a plane, i.e. no cutting edge plane can be defined in the sense of the subject of the present application. Furthermore, it is of importance that the inclination of the flank of a straight cutting edge such as in the device in accordance with DE-AS No. 15 02 100 cannot have any influence on the shape of a cutting arc. The shape of the elliptical cutting arc is determined by the angle .gamma. or, respectively, the rotational cone of the cutting generatrix of the cutting tool, the size of which can be different, depending on the radial distance of the point of contact of the cutting edge in reference to the rotational axis.